Self-Assembly of Any Shape with Constant TileTypes using High Temperature

Cameron Chalk1

Department of Electrical and Computer Engineering, University of Texas - Austin, Austin, TX,USActchalk@utexas.edu

Austin Luchsinger2Department of Computer Science, University of Texas - Rio Grande Valley, Edinburg, TX, USAaustin.luchsinger01@utrgv.edu

Robert Schweller3Department of Computer Science, University of Texas - Rio Grande Valley, Edinburg, TX, USArobert.schweller@utrgv.edu

Tim Wylie4Department of Computer Science, University of Texas - Rio Grande Valley, Edinburg, TX, USAtimothy.wylie@utrgv.edu

AbstractInspired by nature and motivated by a lack of top-down tools for precise nanoscale manufacture,self-assembly is a bottom-up process where simple, unorganized components autonomously combineto form larger more complex structures. Such systems hide rich algorithmic properties—notably,Turing universality—and a self-assembly system can be seen as both the object to be manufacturedas well as the machine controlling the manufacturing process. Thus, a benchmark problem inself-assembly is the unique assembly of shapes: to design a set of simple agents which, based onaggregation rules and random movement, self-assemble into a particular shape and nothing else. Weuse a popular model of self-assembly, the 2-handed or hierarchical tile assembly model, and allow theexistence of repulsive forces, which is a well-studied variant. The technique utilizes a finely-tunedtemperature (the minimum required affinity required for aggregation of separate complexes).

We show that calibrating the temperature and the strength of the aggregation between thetiles, one can encode the shape to be assembled without increasing the number of distinct tiletypes. Precisely, we show one tile set for which the following holds: for any finite connected shapeS, there exists a setting of binding strengths between tiles and a temperature under which thesystem uniquely assembles S at some scale factor. Our tile system only uses one repulsive gluetype and the system is growth-only (it produces no unstable assemblies). The best previous uniqueshape assembly results in tile assembly models use O(

K(S)logK(S)

) distinct tile types, where K(S) isthe Kolmogorov (descriptional) complexity of the shape S.

2012 ACM Subject Classification Theory of computation→Models of computation, Theory of computation→Designand analysis of algorithms→Parallel algorithms→Self-organization, Theory of computation→Designand analysis of algorithms→Distributed algorithms→Self-organization

Keywords and phrases self-assembly, molecular computing, tiling, tile, shapes

Digital Object Identifier 10.4230/LIPIcs.ESA.2018.14

1 This author’s research was supported in part by National Science Foundation Grants CCF-1618895 andCCF-1652824.

2 This author’s research was supported in part by National Science Foundation Grant CCF-1817602.3 This author’s research was supported in part by National Science Foundation Grant CCF-1817602.4 This author’s research was supported in part by National Science Foundation Grant CCF-1817602.

© Cameron Chalk, Austin Luchsinger, Robert Schweller, and Timothy Wylie;licensed under Creative Commons License CC-BY

26th Annual European Symposium on Algorithms (ESA 2018).Editors: Yossi Azar, Hannah Bast, and Grzegorz Herman; Article No. 14; pp. 14:1–14:14

Leibniz International Proceedings in InformaticsSchloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl Publishing, Germany

14:2 Self-Assembly of Any Shape with Constant Tile Types using High Temperature

1 Introduction

Due to the limited tool set for precise fabrication at the nanoscale, the bottom-up approachof self-assembly is an attractive area of research. Such bottom-up approaches, such asDNA origami [17], allow for the assembly of nanoscale materials with detailed, preciselydesigned shapes and patterns. Abstract self-assembly models are used to predict the behaviorof systems wherein simple, separate entities form larger complex structures based on asimple rule set for movement and/or attachment using only local interactions and no globalleader. Such systems include swarm robotics and molecular self-assembly, particularly self-assembling nucleic acid structures such as DNA tiles [9].

A common benchmark in such models, which aims at the manufacture of precise nanoscalestructures, is the self-assembly of shapes. In our case, a shape is defined simply as a finite,connected subset of Z2. The model studied herein is the two-handed tile assembly model(also called the hierarchical tile assembly model) [1]. In this model, the separate entitiesare tiles, adorned with glues. The intuition is that tiles wander about randomly, and whentiles with matching glues meet, the tiles bind to form a larger assembly; further, such largerstructures wander about and may bind to other larger assemblies or tiles.

The main measure of complexity is the number of unique types of tiles necessary andsufficient to uniquely assemble the shape, termed the tile complexity. The temperature ofa system, denoted by τ , is the minimum required binding strength between two entitiesto enforce a stable attachment; the sum of the strengths between the shared glues of twoassemblies must meet or exceed the temperature. Some studies of the model include negative-strength glues [2,8,12–15,20], which are repulsive forces which act against a particular bondbetween two assemblies. Studies of these repulsive forces are motivated by experimentalimplementation of self-assembly systems which exhibit this behavior [16].

Our contributions. We give one tile set with a constant number of distinct tile typeswhich satisfies the following: given any finite connected shape S ⊂ Z2, there exists anassignment of strengths between glues (a glue function) and a temperature τ such that thesystem uniquely assembles S. The system encodes the shape in its temperature parameterτ and its glue function. Then, by utilizing the inclusion of one negative-strength glue type,the system assembles a width-τ assembly. This width-τ assembly is utilized as a seed fora tile set designed by [23] which “runs” the program encoded by the seed to assemble theshape. This work is the first to show that any shape can be built with a constant number ofdistinct tile types (where the glues are a function of τ) at any scale without a staged model5,i.e., it is the first to achieve this in a fully “hands-off” model which requires no experimenterintervention during the assembly process.

Previous results. For self-assembling a shape S, we list the previously known results,which do not use negative glues and use O(1) temperature unless otherwise specified. LetK(S) be the Kolmogorov complexity6 of S, and let T (S) be the (smallest) runtime of aKolmogorov-optimal program outputting S. A tile complexity of Θ

(K(S)

logK(S)

)is known, us-

ing a scale factor of T (S) [23]. With negative-strength glues, a tile complexity of Θ(

K(S)logK(S)

)is known, using a O(1) scale factor [12]. In a staged version of the model, where severalself-assembly systems are run in parallel across a series of bins and then mixed together in

5 In the staged self-assembly model, the results of [3] give a construction which can effectively use O(1) tiletypes to assemble the shape by increasing the number of bins and stages used.

6 The Kolmogorov complexity of S is the number of bits in the smallest program which outputs S w.r.t. auniversal Turing machine. For more information on Kolmogorov complexity, see [11].

C. Chalk, A. Luchsinger, R. Schweller, and T. Wylie 14:3

stages, a tile complexity of Θ(

K(S)logK(S)

)at scale factor T (S) is known for O(1) bins and O(1)

stages along with a method for (optimally) reducing the number of sufficient and necessarytile types by increasing the number of bins and stages [3].

In another staged version of the model, where tiles are partitioned into DNA and RNA types,and RNA types may be “washed away” at a given stage, a tile complexity of Θ

(K(S)

logK(S)

)at

scale factor O(log |S|) is known [7]. In a staged model where the self-assembly process iscontrolled by a chemical reaction network which activates and deactivates tiles’ binding sites,a tile plus reaction network complexity of Θ

(K(S)

logK(S)

)at scale factor O(1) is known [19].

Related work in high-temperature7 self-assembly. The first studies of utilizingtemperature to encode information involved the “online”, mid-assembly-process changing oftemperatures [10, 24]. Our result utilizes a high temperature bonding threshold for self-assembly attachment, which we leverage to encode precise information for guiding the self-assembly process through precisely set glue strengths. A number of recent related works havealso studied the effects of higher temperature self-assembly systems within various models.Within the aTAM, larger temperatures have been shown to affect the possible behaviorof systems [4], and the tile complexity of self-assembled shapes [22]. Within the 2HAM,unique-assembly verification has been shown to be hard for high-temperature systems [21],while the dynamics of certain higher-temperature systems have been shown to be impossibleto simulate at lower temperatures [6].

2 Definitions and ModelIn this section we first define the two-handed tile self-assembly model with both negativeand positive strength glue types. We also formulate the problem of designing a tile assemblysystem that constructs a constant-scaled shape given the optimal description of that shape.

Tiles and Assemblies. A tile is an axis-aligned unit square centered at a point inZ2, where each edge is labeled by a glue selected from a glue set Π. A strength functionstr : Π → Z denotes the strength of each glue. Two tiles equal up to translation have thesame type. A positioned shape is any subset of Z2. A positioned assembly is a set of tilesat unique coordinates in Z2, and the positioned shape of a positioned assembly A is the setof those coordinates. For a given positioned assembly Υ, define the bond graph GΥ to bethe weighted grid graph in which each element of Υ is a vertex and the weight of an edgebetween tiles is the strength of the matching coincident glues or 0.8 A positioned assemblyC is τ -stable for positive integer τ provided the bond graph GC has min-cut at least τ .

For a positioned assembly A and integer vector ~v = (v1, v2), let A~v denote the positionedassembly obtained by translating each tile in A by vector ~v. An assembly is a translation-free version of a positioned assembly, formally defined to be a set of all translations A~v ofa positioned assembly A. An assembly is τ -stable if and only if its positioned elements areτ -stable. A shape is the set of all integer translations for some subset of Z2, and the shape ofan assembly A is defined to be the set of the positioned shapes of all positioned assembliesin A. The size of either an assembly or shape X, denoted as |X|, refers to the number ofelements of any positioned element of X.

7 We say high-temperature self-assembly for consistency with previous literature. The term high tempera-ture may be misleading; e.g., we are not attempting to model what happens in DNA-based self-assemblysystems when the literal temperature of the system is raised to high values. Intuitively, higher temperaturein this model implies more fine-grained glue strengths. Another natural way to think of high temperatureis to fix the temperature to one, but allow rational glue strengths.

8 Note that only matching glues of the same type contribute a non-zero weight, whereas non-equal gluesalways contribute zero weight to the bond graph. Relaxing this restriction has been considered [5].

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14:4 Self-Assembly of Any Shape with Constant Tile Types using High Temperature

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Figure 1 The black lines between two tiles indicate unique unimportant τ -strength bonds. Ifτ = 2, str(A) = 1 and str(B) = 1, then the two assemblies in (a) are τ -combinable, since str(A) +

str(B) ≥ τ and the positioned assemblies may be translated such that the A and B glues arealigned— such a combination is termed cooperative binding, since neither the A nor B glue are alonesufficient to satisfy τ -combination. In (b), we consider two cases concerning the negative strengthglue X. If τ = 2, str(C) = 2, and str(X) = −1, then the assemblies in (b) are not τ -combinablesince str(C) + str(X) < τ . If τ = 1, str(C) = 2, str(D) = 2, str(E) = 1 and str(X) = −1,then the assemblies are τ -combinable since str(C) + str(X) ≥ τ ; however, the resultant assemblyis unstable, since a cut along the X and E glue has strength str(X) + str(E) < τ ; this violates thevalid growth-only system definition.

Combinable Assemblies. Two assemblies are τ -combinable provided they may attachalong a border whose strength sums to at least τ . Formally, two assemblies A and B areτ -combinable into an assembly C provided GC′ for any C ′ ∈ C has a cut (A′, B′) of strengthat least τ for some A′ ∈ A and B′ ∈ B. We call C a combination of A and B. Figure 1gives examples of combinable and not combinable assemblies.

2.0.0.1 Two Handed Assembly Model: Growth-only Version

A two-handed tile assembly system (2HAM system) is an ordered pair (T, τ) where T is a setof single tile assemblies, called the tile set, and τ ∈ N is the temperature. In the growth-onlymodel, assembly proceeds by repeated combination of assembly pairs to form new assembliesstarting from the initial tile set. The producible assemblies are those constructed in this way.

I Definition 1 (2HAM Producibility (growth-only)). For a given 2HAM system Γ = (T, τ),the set of producible assemblies of Γ, denoted PRODΓ, is defined recursively:

(Base) T ⊆ PRODΓ

(Combinations) For any A,B ∈ PRODΓ such that A and B are τ -combinable into C, thenC ∈ PRODΓ.The inclusion of negative glues, in general, allows for unstable assemblies to be pro-

ducible. In previous literature, such assemblies “detach”, forming two new producible as-semblies. We impose the following constraint on growth-only systems which disallows pro-duction of unstable assemblies which would fall apart. Satisfying the growth-only constraintargues that the system has simpler kinetics than a non-growth-only system since the systemdoes not rely on detachment events.

For a system Γ = (T, τ), we say A →Γ1 B for assemblies A and B if A is τ -combinable

with some producible assembly to yield B, or if A = B. Intuitively this means that A maygrow into assembly B through one or fewer combination. We define the relation →Γ to bethe transitive closure of→Γ

1 , ie., A→Γ B means that A may grow into B through a sequenceof combinations.

I Definition 2 (Valid Growth-Only System). A 2HAM system Γ = (T, τ) is a valid growth-only system if for all A ∈ PRODΓ, A is τ -stable.

I Definition 3 (Terminal Assemblies). A terminal assembly of a valid growth-only 2HAMsystem is a producible assembly that cannot combine with any other producible assembly.Formally, an assembly A ∈ PRODΓ of a 2HAM system Γ = (T, τ) is terminal provided A isnot τ -combinable with any producible assembly of Γ.

We formalize what it means for a 2HAM system to uniquely build a given assembly ora given shape.

C. Chalk, A. Luchsinger, R. Schweller, and T. Wylie 14:5

Figure 2 A simplified overview of the growing step. Si is a width-Θ(i), height-Θ(2i) assemblywith particular exposed edge glues. Si nondeterministically assembles one of two assemblies; a topand bottom. The top and bottom share one glue of strength 2τ − 1 shown in yellow, and i many−1 strength glues shown in red. Thus, the top and bottom bind with strength 2τ − i− 1, which isτ -stable only if i < τ . The resultant assembly adds two width and doubles the height of Si, so itsdimensions are Θ(i+ 1)×Θ(2i+1). Further, its exposed glues allow the process to repeat.

I Definition 4 (Unique Assembly). A 2HAM system uniquely produces an assembly A ifall producible assemblies have a growth path towards the terminal assembly A. Formally,a 2HAM system Γ = (T, τ) uniquely produces an assembly A provided that A is terminal,and for all B ∈ PRODΓ, B →Γ A.

I Definition 5 (Unique Shape Assembly9). A 2HAM system uniquely produces a shape S ifall producible assemblies have a growth path to a terminal assembly of shape S. Formally,a 2HAM system Γ = (T, τ) uniquely assembles a finite shape S if for every A ∈ PRODΓ, thereexists a terminal A′ ∈ PRODΓ of shape S such that A→Γ A′.

3 Assembly of General Shapes with Constant Tiles

Here we give the main construction of the paper. First presented is our key contribution—assembly of a precise-width rectangle— detailed in Subsection 3.1, followed by its composi-tion with established techniques from [23] for the main result in Subsection 3.2.

3.1 Key idea: precise-width rectangle using O(1) tile types

Here, we present a construction for building a precise-width rectangle from a constant-bounded set of tile types. Note that the convention in this paper is width × height. Formally,

I Lemma 6. Given a temperature τ > 2, there exists a negative glue, growth-only 2HAMtile system Γ = {T, τ} such that |T | = O(1) and Γ uniquely produces an assembly which isa (18 + 4τ)× (2τ+5 − 6) rectangle.

I Proof 1. We give a proof by construction. Unless explicitly stated otherwise, all glueshave strength d τ2 e, so at least two matching glues are required for a τ -stable attachment.This is called cooperative binding. The construction is split into two steps: growing andfinishing. Figure 2 shows a simplified overview of the growing step. The growing step ofthe construction concerns producing an assembly with width 7 + 2τ , through a processconsisting of τ iterations of growth, each adding a constant-bounded width to the assembly.

9 Some previous literature calls this strict self-assembly, typically to contrast another definition, weak self-assembly; we choose the name unique shape assembly to contrast unique assembly.

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Figure 3 The base assembly, shown in two separate subassemblies; (a) shows the top subassembly,and (b) the bottom. The two subassemblies combine using cooperative binding at the strength-d τ

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glues labeled X. The dotted line indicates distinct tile types which attach along the path with fullτ strength glues. The snaking pattern ensures that each subassembly is complete before both X

glues are available. Once these two subassemblies bind, the resultant assembly satisfies S0.

The i + 1th iteration of growth is initiated by a combination of two assemblies with totalbinding strength 2τ − i − 1; thus, after the τ th iteration of growth, the binding strengthwhich would initiate the next iteration of growth has total binding strength 2τ − τ − 1 < τ ,and growth halts.

The finishing step involves the system’s “detecting” that the growth process has com-pleted. This is achieved using the following technique: by adding a total strength of 1 sharedbetween two assemblies at each iteration of growth, once the assemblies have completed τrepetitions of growth, they bind with strength τ . This step also gives the system its propertyof unique assembly of an assembly whose shape is a rectangle (and not just unique shapeassembly). That is, there is exactly one terminal assembly of the system— as opposed toseveral terminal assemblies with the correct rectangular shape. Maintaining this propertyin this lemma is required to achieve the same property in Theorem 7.

3.1.1 Growing

The construction is described and proven correct via induction. The induction is on iter-ations of rectangular assemblies with well-defined exposed glues, termed Si. Formally, Sirefers to a (7 + 2i) × (2i+4 − 6) rectangular assembly with the following exposed glue la-bels, written as strings built by concatenating the glue labels in left-to-right/top-to-bottomorder):

North glues: qpigni+5

East glues: E2i+3−5R2E2i+3−3

South glues: QP iGN i+5

West glues: LW 2i+3−7LW 4LW 2i+3−7L

The goal is to show that if an assembly satisfying Si is producible, then an assembly satisfyingSi+1 is producible iff i < τ . Further, only O(1) tile types are used in the inductive step.Then it suffices to show that S0 is producible in O(1) tile types, implying Sτ is producible,which has width × height as in the lemma statement.

Base case. An assembly satisfying S0 is shown to be trivially assembled by a set ofO(1) tile types in Figure 3.

Inductive step. The next three paragraphs describe the inductive step. The goal isto show that if Si is producible, then a top and bottom assembly are producible which can

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Figure 4 An assembly satisfying Si. The dots indicate an omitted set of repeated tiles; e.g.,the dots between the tiles exposing p indicate the omission of the i glues with label p. On theright are the keystones, which attach cooperatively using R glues. Only one keystone may attach,introducing nondeterminism; this is how the producibility of two assemblies, a top and bottomassembly, are implied by the production of one assembly satisfying Si.

bind to produce Si+1 iff i < τ . Consider an assembly with exposed glues satisfying Si. Thetwo R glues exposed allow attachment of a keystone assembly via cooperative binding asseen in Figure 4. There are two keystone types: up and down. Only one may attach. TheL glues in the middle of the west-side exposed glues, spaced by four W glues, also allow theattachment of a supertile using cooperative binding. Once the keystone or west-side supertilehas attached, a single tile type suffices to attach tiles along any long set of repeating glueson the assembly (e.g., the E glues on the east side), until the glue is no longer available,as seen in Figure 5. This type of single tile type attaching along arbitrary walls is termedpropagation of a tile.

The tiles which propagate to the corners of the west side of the assembly allow the at-tachment of three tiles around the corner which allow propagation of tiles along the northand south faces of the assembly. Once these tiles propagate, depending on which keystonewas attached to the assembly, the attachment of a tooth occurs on the corresponding face(e.g., on the north face if the up keystone was attached) as shown in Figure 6. The toothis a supertile with a specific geometry which will be motivated later on. The tooth initiatesa propagation of tiles along the corresponding face. The result is the production of twoassemblies, one having attached the up keystone and attached all previously discussed prop-agating tiles and supertiles, and one having attached the down keystone and similar tiles.We refer to the former as a top assembly, and the latter as a bottom assembly.

When a top assembly and bottom assembly attach, the result is an assembly satisfyingSi+1. A top assembly and bottom assembly are designed to attach iff i < τ . When i ≥ τ , thebinding strength between a top and bottom assembly is τ − 1, and thus is insufficient. Thisdesign can be seen in Figure 7. Note that the −1 glues are propagated via the N -labeledglues from the base assembly. Since Si has i + 5 many N glues exposed, 5 of which are

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Figure 5 Once the keystone and supertile on the left have attached, a sequence of single tileattachments may occur using cooperative binding at newly available glues. The tiles are designedsuch that they may attach along arbitrarily long faces, as long as the appropriate glue is exposed(e.g., a W glue in the top-left tile’s case).

covered by the tooth, the bottom/top assembly which assembles from Si exposes i many −1

glues. Then the bonding strength between top and bottom assemblies is 2τ − 1 − i, whichis less than τ iff i ≥ τ . The complementary geometry of the teeth ensure that a top andbottom assembly which are assembled from Si and Sj respectively, with i 6= j, cannot aligntheir a glues and will not attach.

Dimensions of Si. The base assembly satisfying S0 is 7× 10. When a top and bottomassembly attach, both have added 2 width in tiles; one tile width propagated on the westside, and one on the east. Then the width of Si is 7 + 2i. A top and bottom assembly whichhave combined add 6 height in tiles on top of doubling in height: 2 via tile propagationon the top and bottom, and 4 along where the top and bottom assemblies attach. Thenthe height of Si, h(i), is defined by the recurrence h(i) = 2h(i − 1) + 6 with h(0) = 10.Solving the recurrence gives a height of 2i+4− 6. Then consider a combination of a top andbottom assembly which formed from some assembly satisfying Si−1; the resultant dimensionis (7 + 2i)× (2i+4 − 6).

3.1.2 Finishing

When a top and bottom assembly combine to form an assembly satisfying Sτ , the growingstep shows that the process will not continue to produce Sτ+1. However, the attachmentof a supertile on the west side, a keystone, and the resultant tile propagation still occurs.The teeth attach and so do the tiles which propagate resulting from the attachment of a

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Figure 6 At the top-right and bottom-right corners, tiles attach which indicate that the left-handside tile propagation has reached the right-hand side of the assembly. In the case of an assemblywhich has attached an up keystone, the tooth attaches on the top side of the assembly, and initiatesa propagation of tiles along the top face. A tooth with complementary geometry will attach on thebottom side of the assembly if a down keystone attaches instead, as can be seen in Figure 7.

tooth. Then an assembly satisfying Sτ implies the production of the corresponding top andbottom assemblies. These assemblies are not rectangular. This step involves detecting thatthe iterative process has reached τ repetitions, and the system should finish its rectangle.

Figure 8 gives an overview of the finishing step. The technique discussed in the growingphase is employed by two disjoint tile sets, one called system 1 and the other system 2. Thesets of glues on the tiles in the two systems are disjoint except for two glues: the −1 glue, andthe +2 glue described but not used in the growing phase. In the growing phase, recall that onthe north face of a bottom assembly of system 1 which assembled from Si, there are i manystrength 2 glues labeled +2 exposed which are not used (recall Figure 7). These glues aredesigned to match with the corresponding glues in system 2. Then the strength of bindingbetween these shared glues is 2i − i = i. Thus, only when i ≥ τ is this binding τ -stable.

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Figure 7 A bottom assembly (left) and top assembly (right). At the top of the figure are the tileswhose glues bond a bottom and top assembly; in particular, the a and −1 glues, with str(a) = 2τ−1

and str(−1) = −1. A top and bottom assembly grown from an assembly Si each expose i glueswith strength −1. Then the strength of the attachment between them is 2τ − i− 1, and is sufficientwhen i < τ but insufficient when i = τ . Note that +2 and +2′ glues do not match; their purposeis described later in the finishing step.

Similarly, system 1’s top assembly attaches with the system 2’s bottom assembly underthe same constraint. These resultant assemblies expose cooperative binding locations whichwere not present before this attachment, allowing these two new assemblies to combine,and then fill into a rectangle using a O(1)-sized tile set. Next, we give the dimensions ofthe completed rectangular assembly: system 1’s top and system 2’s bottom assembly, onceattached, form a (7 + 2(τ + 1)) = (9 + 2τ)× (2(τ+1)+4− 6) = (2τ+5− 6) assembly— this canbe derived from the combination of two assemblies satisfying Sτ assembling into an assemblysatisfying Sτ+1 not in exposed glues, but in size. System 1’s top and system 2’s bottomare combined with system 1’s bottom and system 2’s top into one via a width-two column,resulting in a 2(9 + 2τ) + 2 = 18 + 4τ × 2τ+5 − 6 assembly. J

3.2 From rectangle to shapeTo assemble the target shape, a technique is combined with Lemma 6. The technique,shown by Soloveichik and Winfree [23], first assembles a seed block bearing a representation(a series of exposed glues) of a Kolmogorov-optimal program which computes the spanningtree of S. A O(1)-sized tile set is used to “run” the program (via Turing machine (TM)simulation) from the seed block. The seed block assembles into a c × c assembly, logicallyrepresenting one coordinate of S. Assembly proceeds from the seed block in a subset of thefour cardinal directions depending on the spanning tree computed by the program. If thespanning tree has an edge in a direction to a coordinate adjacent to the seed block, a c× cgrowth block is assembled in that direction. Each time a growth block is assembled, theprogram is run again to determine which adjacent coordinates (w.r.t. the growth block) are

C. Chalk, A. Luchsinger, R. Schweller, and T. Wylie 14:11

System 1 Top

System 2Bottom

System 2 Top

System 1 Bottom

Figure 8 An overview of the finishing step. The system described in the growing step, denotedhere as system 1, is repeated, denoted system 2, such that the only matching glues between thetwo systems are a strength-2 glue type and the one strength-(−1) glue type. Between a system 1top/bottom and system 2 bottom/top assembled from Si, the number of shared strength-2 gluesand strength-(−1) glues is i, so the sum of strengths between shared glues is 2i− i = i. This allowsthe top/bottom assembly of system 1 to make a τ -stable attachment to the bottom/top of system2 only after each system assembles Sτ , thus detecting when the rectangle has Θ(τ) width. Oncethese tops and bottoms attach, new cooperative binding locations initiate a constant-sized set oftiles to bind the two rectangles, simply to satisfy unique assembly of the rectangle.

connected by edges in the spanning tree; if so, a growth block is assembled in that direction.After assembling all growth blocks, the unique assembly is a c× c scaled version of S.

The seed block of [23] is assembled using O( K(S)logK(S) ) tile types. In our case, the seed

block is assembled using the O(1)-sized tile set of Lemma 6: we assemble a rectangle ofwidth n where n is the length of a unary encoding of the Kolmogorov-optimal program.This seed is combined with the O(1)-sized tile set which runs the program and assemblesthe shape. Figure 9 is a simplified overview of constructing a seed block compatible with aTM simulating tileset. The formal result is as follows:

I Theorem 7. Given a shape S, there exists a negative glue, growth-only 2HAM tile systemΓ = {T, τ} with |T | = O(1) whose unique assembly has shape S at some scale factor.

I Proof 2. The system is a union of the Lemma 6 system and a subset of the TM simulatingtile set of [23]. If the entire TM simulating tile set is added to the system, depending onthe seed block, some tiles may never bind to the seed block, and thus do not grow intothe target shape and violate unique assembly definition. We include the subset of the TMsimulating tile set which will be used by the program encoded by the seed block. Observethat the unique assembly produced by the system of Lemma 6 is not a square, nor doesit have any exposed glues designed to bind with the TM simulation tile set. In order toassemble a square from the terminal assembly of Lemma 6, two such rectangular assembliesare assembled in parallel, one which is rotated 90 degrees— this “rotation” is w.r.t. theother rectangular assembly and the way the two assemblies will bind. These two assembliescombine and then “fill-in” to a square trivially using a constant-sized tile set— similar topropagation of tiles along an edge, cooperative binding can be used to add tiles between twoassembled rectangles to assemble a square (technique first used in [18]). Once assembly ofthe square is complete, more tile propagation via another constant-sized tile set assembles aone-tile perimeter which exposes glues— described in the following paragraph— which allowthe assembly to act as a seed block similar to the Soloveichik and Winfree construction [23].

Let P be the (binary) program used to assemble S via the construction of [23], R besome mapping from binary strings to unary strings, and R′ be some mapping from unarystrings {1}i with i ∈ N to unary strings {1}j with j = 20 + 4m for m ∈ N— the intuition forthe mapping R′ is to map arbitrary unary strings to numbers which are widths of assembliesassembled by Lemma 6.

ESA 2018

14:12 Self-Assembly of Any Shape with Constant Tile Types using High Temperature

Figure 9 The combination of two Lemma 6 constructions into a seed block. The two-tile assemblyin the first subfigure initializes the attachment of the set of white tiles, which indicate a constant-sized set of filler tiles which are used to fill in a full square. Once the square is filled in, newcooperative binding locations are exposed where the filler tiles meet the non-filler tiles. At thislocation, tiles begin to propagate, adding a one-tile perimeter to the assembly. The orange tiles onthe outmost perimeter of the rightmost figure demarcate the beginning and ending of glues exposingthe unary program which constructs the shape S via the TM simulation of [23]. The rest of theperimeter exposes glues which the TM simulation ignores.

Once the square seed block is assembled and a one-tile perimeter is attached, three gluetypes are exposed: 1, b, and λ. Across the length of filler tiles (those in the square which arenot from the Lemma 6 construction), λ glues are placed; these symbols are ignored by theTM simulating tile set. The b glues are placed at the beginnings and ends of the edges of thesquare; these indicate where to start the TM simulation. Along the edges of the rectanglesfrom the Lemma 6 construction, glues labeled 1 are placed; these 1 glues are the relevantglues logically. The TM simulation converts these to unary strings by R′−1, and then to Pby R−1. Then the TM simulation tiles run the program P which assembles the shape. J

4 Future Work

The most apparent direction for future work is to achieve the unique assembly of shapes at aO(1) scale factor with O(1) tile types. This result may be achieved through a combination ofthe techniques used in this work and in [12], which achievesO(1)-scale factor withO( K(S)

logK(S) )

tile types where K(S) is the Kolmogorov complexity of the shape S. Their constructionutilizes a dynamic behavior of negative glues not utilized in this work called “breaking”:the combination of two assemblies may result in an unstable assembly, which then breaksinto two assemblies— a formal model and some usages of breakage may be seen in [2,8,20].Their usage of breaking involves performing a computation— via a self-assembly processwhich simulates a TM— which builds the shape pixel-by-pixel (using O(1)-sized assembliesper pixel), and then breaks the TM simulating assembly into O(1)-sized pieces, leaving theshape S at a O(1) scale factor (along with “small garbage” of O(1) size). That techniquemay be applicable to the construction given in this work in order to break the precise-widthrectangles after they are used as input for the TM which outputs S.

Another direction might be to achieve the unique assembly of scaled shapes with O(1)

tile types using only positive-strength glues. We have briefly discussed previous positive-strength results which use Θ

(K(S)

logK(S)

)tile types. Could this be lowered to O(1) tile types

by calibrating the temperature and glue strengths, or is there some super-constant lowerbound that cannot be breached?

C. Chalk, A. Luchsinger, R. Schweller, and T. Wylie 14:13

References

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